Assess graduate students' understanding of modes of convergence in L^p spaces (a.e., in measure, in L^p, and weak convergence): require clear definitions, proofs of standard implications (e.g. L^p ⇒ in measure, in measure ⇒ subsequence a.e.), identification of necessary/sufficient conditions (e.g. dominated convergence, Vitali’s theorem/uniform integrability), analysis of weak vs. strong convergence (including role of reflexivity), and construction of counterexamples showing failure of converses. Emphasize rigorous argumentation and common illustrative examples.